Fast and Simple Parallel Wavelet Tree and Matrix Construction

The wavelet tree (Grossi et al. [SODA, 2003]) and wavelet matrix (Claude et al. [Inf. Syst., 47:15--32, 2015]) are compact indices for texts over an alphabet $[0,\sigma)$ that support rank, select and access queries in $O(\lg \sigma)$ time. We first present new practical sequential and parallel algorithms for wavelet matrix construction. Their unifying characteristics is that they construct the wavelet matrix bottom-up, i.e., they compute the last level first. We also show that this bottom-up construction can easily be adapted to wavelet trees. In practice, our best sequential algorithm is up to twice as fast as the currently fastest sequential construction algorithm (serialWT), simultaneously saving a factor of 2 in space. On 4 cores, our best parallel algorithm is at least twice as fast as the currently fastest parallel algorithm (recWT), while also using less space. This scales up to 32 cores, where we are about equally fast as recWT, but still use only about 75% of the space. An additional theoretical result shows how to adapt any wavelet tree construction algorithm to the wavelet matrix in the same (asymptotic) time, using only little extra space.